diagonalens

In geometry, a diagonal is a line segment that connects two nonconsecutive vertices of a polygon. In three dimensions, a diagonal can connect two nonadjacent vertices of a polyhedron; diagonals lying on a single face are called face diagonals, while those that pass through the interior are space diagonals.

For a polygon with n vertices, the number of diagonals is n(n-3)/2. This counts all pairs of vertices that are not adjacent. In convex polygons, all diagonals lie inside the figure; in concave polygons, some diagonals may lie outside the boundary.

Diagonals can intersect inside a polygon, and their arrangement is central to procedures such as triangulation, where a polygon is divided into nonoverlapping triangles using diagonals.

In a rectangle, the two diagonals are equal in length and bisect each other; they are not perpendicular unless the rectangle is a square. Diagonals also play a role in higher-dimensional figures: in a rectangular prism, the space diagonal—the line segment joining opposite vertices—has length sqrt(a^2 + b^2 + c^2) if the edge lengths are a, b, and c.

In linear algebra, the diagonal of a square matrix consists of the elements a_ii (the main diagonal). The sum of these elements is called the trace of the matrix, and diagonals are central to concepts such as diagonalization and eigenvalues.

Diagonals have broad applications in geometry, design, computer graphics, and proofs, where they help describe shape, symmetry, and spatial relationships.